How to find the likelihood of engineering failure in next 5 years using data and statistical method and Give example in

# -------------------------------------------------------------
# 1. Setup & Synthetic Data Generation
# -------------------------------------------------------------
set.seed(42)

# True underlying system parameters:
# Shape (beta) > 1 indicates wear-out over time
true_shape <- 2.5
true_scale <- 10.0   # Characteristic life in years

# Simulate historical failure records (50 observed failures)
n_samples <- 50
historical_ttf <- rweibull(n_samples, shape = true_shape, scale = true_scale)

# -------------------------------------------------------------
# 2. Fit Weibull Model via Maximum Likelihood Estimation (MLE)
# -------------------------------------------------------------
# Fit a Weibull distribution using base R's optim()
log_lik <- function(params, data) {
  shape <- params[1]
  scale <- params[2]
  if (shape <= 0 || scale <= 0) return(Inf)
  -sum(dweibull(data, shape = shape, scale = scale, log = TRUE))
}

fit <- optim(par = c(1, mean(historical_ttf)), 
             fn = log_lik, 
             data = historical_ttf, 
             hessian = TRUE)

est_shape <- fit$par[1]
est_scale <- fit$par[2]

cat("Fitted Parameters:\n")
cat(sprintf("  Estimated Shape (beta): %.3f\n", est_shape))
cat(sprintf("  Estimated Scale (eta):  %.3f years\n\n", est_scale))

# -------------------------------------------------------------
# 3. Monte Carlo Simulation for Next 5-Year Horizon
# -------------------------------------------------------------
# Scenario: An existing machine has already operated for t_current years.
# We want to know the probability it fails within the next delta_t years.

t_current <- 3.0   # Current age in years
delta_t   <- 5.0   # Forecast window (next 5 years)
t_horizon <- t_current + delta_t

n_sims <- 100000   # Number of Monte Carlo iterations

# Step A: Estimate parameter uncertainty using the Fisher Information Matrix
var_covar <- solve(fit$hessian)
sim_params <- MASS::mvrnorm(n_sims, mu = c(est_shape, est_scale), Sigma = var_covar)

# Filter out non-positive samples if any occur
valid_idx  <- sim_params[, 1] > 0 & sim_params[, 2] > 0
sim_params <- sim_params[valid_idx, ]
n_sims_valid <- nrow(sim_params)

# Step B: Simulate conditional time-to-failure
# Using inverse transform sampling conditioned on T > t_current:
# u ~ Uniform(S(t_current), 1) where S(t) = exp(-(t/scale)^shape)
u <- runif(n_sims_valid)
s_current <- exp(- (t_current / sim_params[, 2])^sim_params[, 1])

# Generate remaining lifetime conditioned on survival to t_current
u_cond <- u * s_current
sim_ttf <- sim_params[, 2] * (-log(u_cond))^(1 / sim_params[, 1])

# Step C: Count failures within the next 5 years (i.e., TTF <= t_current + delta_t)
failures_in_5_years <- sim_ttf <= t_horizon
prob_failure_mc <- mean(failures_in_5_years)

# 95% Confidence Interval for the Monte Carlo estimate
ci_mc <- quantile(as.numeric(failures_in_5_years), probs = c(0.025, 0.975))
se_mc <- sqrt((prob_failure_mc * (1 - prob_failure_mc)) / n_sims_valid)

cat("Monte Carlo Forecast Results:\n")
cat(sprintf("  Probability of failure in next 5 years: %.2f%%\n", prob_failure_mc * 100))
cat(sprintf("  95%% Confidence Interval: [%.2f%%, %.2f%%]\n\n", 
            (prob_failure_mc - 1.96 * se_mc) * 100, 
            (prob_failure_mc + 1.96 * se_mc) * 100))

# -------------------------------------------------------------
# 4. Analytical Verification
# -------------------------------------------------------------
# Theoretical conditional failure probability for comparison:
p_analytic <- (pweibull(t_horizon, est_shape, est_scale) - pweibull(t_current, est_shape, est_scale)) / 
              (1 - pweibull(t_current, est_shape, est_scale))

cat(sprintf("Analytical conditional probability: %.2f%%\n", p_analytic * 100))